| L(s) = 1 | + 2·2-s + 3-s + 2·4-s + 5-s + 2·6-s + 3·7-s − 2·9-s + 2·10-s + 2·12-s + 6·13-s + 6·14-s + 15-s − 4·16-s − 4·17-s − 4·18-s − 19-s + 2·20-s + 3·21-s − 3·23-s + 25-s + 12·26-s − 5·27-s + 6·28-s − 3·29-s + 2·30-s + 7·31-s − 8·32-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 0.577·3-s + 4-s + 0.447·5-s + 0.816·6-s + 1.13·7-s − 2/3·9-s + 0.632·10-s + 0.577·12-s + 1.66·13-s + 1.60·14-s + 0.258·15-s − 16-s − 0.970·17-s − 0.942·18-s − 0.229·19-s + 0.447·20-s + 0.654·21-s − 0.625·23-s + 1/5·25-s + 2.35·26-s − 0.962·27-s + 1.13·28-s − 0.557·29-s + 0.365·30-s + 1.25·31-s − 1.41·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 755 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 755 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.129042573\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.129042573\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | Isogeny Class over $\mathbf{F}_p$ |
|---|
| bad | 5 | \( 1 - T \) | |
| 151 | \( 1 + T \) | |
| good | 2 | \( 1 - p T + p T^{2} \) | 1.2.ac |
| 3 | \( 1 - T + p T^{2} \) | 1.3.ab |
| 7 | \( 1 - 3 T + p T^{2} \) | 1.7.ad |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 - 6 T + p T^{2} \) | 1.13.ag |
| 17 | \( 1 + 4 T + p T^{2} \) | 1.17.e |
| 19 | \( 1 + T + p T^{2} \) | 1.19.b |
| 23 | \( 1 + 3 T + p T^{2} \) | 1.23.d |
| 29 | \( 1 + 3 T + p T^{2} \) | 1.29.d |
| 31 | \( 1 - 7 T + p T^{2} \) | 1.31.ah |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 - 8 T + p T^{2} \) | 1.41.ai |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 + 10 T + p T^{2} \) | 1.53.k |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 + p T^{2} \) | 1.61.a |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 + 7 T + p T^{2} \) | 1.73.h |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 + 4 T + p T^{2} \) | 1.89.e |
| 97 | \( 1 - 16 T + p T^{2} \) | 1.97.aq |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.79804710749239572112210800926, −9.255688914203212049107356050900, −8.615447624830345390354725416478, −7.82556193345396651220452366345, −6.34482587049913083822378109346, −5.87830109403687891950778211108, −4.78076315187482584125392594040, −3.98936712639549159890754077028, −2.91156616338194989784912694440, −1.83522504840789766101819691778,
1.83522504840789766101819691778, 2.91156616338194989784912694440, 3.98936712639549159890754077028, 4.78076315187482584125392594040, 5.87830109403687891950778211108, 6.34482587049913083822378109346, 7.82556193345396651220452366345, 8.615447624830345390354725416478, 9.255688914203212049107356050900, 10.79804710749239572112210800926
